Geometric morphism

Remarks

Example: let $g \colon X \to Y$ be a continuous map of topological spaces. The direct image $g_* F(V) = F(g^{-1}(V))$ and the inverse image $g^*$ of sheaves form a geometric morphism $\mathrm{Sh}(X) \to \mathrm{Sh}(Y)$. The inverse image preserves finite limits, because stalks of $g^*G$ are stalks of $G$ and finite limits of sheaves are computed stalkwise. Non-example: on $\mathbf{Set}$, take $f^* = - \times A$ and $f_* = (-)^A$ for a set $A$ with two elements. Then $f^* \dashv f_*$, but $f^*$ sends the terminal set $1$ to $A$, which is not terminal, so clause 2 fails. For a topos $\mathcal{E}$, a point of $\mathcal{E}$ is a geometric morphism $\mathbf{Set} \to \mathcal{E}$. Every Grothendieck topos has, up to isomorphism, exactly one geometric morphism to $\mathbf{Set}$.